Linear Theory of Ionization Cooling in 6 D

Linear Theory of Ionization Cooling in 6 D Kwang-Je Kim & Chun-xi Wang University of Chicago and Argonne National Laboratory Cooling Theory/Simulation Day Illinois Institute of Technology February 5, 2002 KJK 2/5/02 IIT Cooling Theory/Simulation Day 1 Advanced Photon Source

• Theory development. . . . . Kim Kwang-Je • Examples and asymmetric beams. . . . Wang Chun-xi KJK 2/5/02 IIT Cooling Theory/Simulation Day 2 Advanced Photon Source

Ionization Cooling Theory in Linear Approximation • Similar in principle to radiation damping in electron storage rings, but needs to take into account: - Solenoidal focusing and angular momentum - Emittance exchange • Slow evolution near equilibrium can be described by five Hamiltonian invariants KJK 2/5/02 IIT Cooling Theory/Simulation Day 3 Advanced Photon Source

Equation of Motion • Phase space vector • • KJK 2/5/02 IIT Cooling Theory/Simulation Day 4 Advanced Photon Source

Emittance Exchange KJK 2/5/02 IIT Cooling Theory/Simulation Day 5 Advanced Photon Source

Hamiltonian Under Consideration Solenoid + dipole + quadrupole + RF + absorber Goal: theoretical framework and possible solution Lab frame solenoid dipole quadrupole r. f. rotating frame with symmetric focusing , KJK 2/5/02 IIT Cooling Theory/Simulation Day 6 Advanced Photon Source

Equations for Dispersion Functions In Larmor frame Dispersion function decouples the betatron motion and dispersive effect KJK 2/5/02 IIT Cooling Theory/Simulation Day 7 Advanced Photon Source

Coordinate Transformation • Rotating (Larmor) frame • Decouple the transverse and longitudinal motion via dispersion: x = xb + Dx , Px = Pxb + Dx • Dispersion vanishes at rf KJK 2/5/02 IIT Cooling Theory/Simulation Day 8 Advanced Photon Source

Wedge Absorbers qw KJK 2/5/02 IIT Cooling Theory/Simulation Day 9 Advanced Photon Source

Natural ionization energy loss is insufficient for longitudinal cooling Transverse cooling momentum gain momentum loss net loss slope is too gentle for effective longitudinal cooling Will be neglected KJK 2/5/02 IIT Cooling Theory/Simulation Day 10 Advanced Photon Source

Model for Ionization Process in Larmor Frame Transverse: M. S. Longitudinal: wedge : Average loss replenished by RF KJK 2/5/02 IIT Cooling Theory/Simulation Day straggling 11 Advanced Photon Source

Equation for 6 -D Phase Space Variables • • x = x + Dx, Px = Px + z = z Dispersion vanishes at cavities Drop suffix KJK 2/5/02 IIT Cooling Theory/Simulation Day 12 Advanced Photon Source

Equilibrium Distribution • Linear stochastic equation Gaussian distribution • For weak dissipation, the equilibrium distribution evolves approximately as Hamiltonian system. I is a quadratic invariant with periodic coefficients. KJK 2/5/02 IIT Cooling Theory/Simulation Day 13 Advanced Photon Source

Quadratic Invariants • Three Courant-Snyder invariants: ( , , ), ( z, z); Twist parameters for and || • Two more invariants when x = y: These are complete set! KJK 2/5/02 IIT Cooling Theory/Simulation Day 14 Advanced Photon Source

Beam Invariants, Distribution, and Moments • Beam invariants (emittances): • Distribution: KJK 2/5/02 IIT Cooling Theory/Simulation Day 15 Advanced Photon Source

Beam Invariants, Distribution, and Moments (contd. ) • Non-vanishing moments: (b) These are the inverses of Eq. (a). KJK 2/5/02 IIT Cooling Theory/Simulation Day 16 Advanced Photon Source

Evolution Near Equilibrium • i are slowly varying functions of s. • • • Insert • Use Eq. (b) to convert to emittances. KJK 2/5/02 IIT Cooling Theory/Simulation Day 17 Advanced Photon Source

Evolution Near Equilibrium (contd. ) • Diffusive part: straggling and multiple scattering . x(s+Ds) = x(s)-Dx . Px(s+Ds) = Px(s)-Dx + < > = < > = 0 < 2> = Ds, < > = 0 KJK 2/5/02 IIT Cooling Theory/Simulation Day 18 Advanced Photon Source

Emittance Evolution Near Equilibrium s = -( -ec-) s+ec+ a+es+ xy+b L+ s, a = -( -ec-) a+ec+ s+ a, xy = -( -ec-) xy+es+ s+ xy, L = -( -ec-) L+b s+ L, z = -( +2 ec-) z+ z, C± = cos(q. D-qw), s± = sin(q. D ± qw), s- = sin (q. D -qw) b = xb + aes- + be s- KJK 2/5/02 IIT Cooling Theory/Simulation Day 19 Advanced Photon Source

The Excitations KJK 2/5/02 IIT Cooling Theory/Simulation Day 20 Advanced Photon Source

Remarks • Reproduces the straight channel results for D = 0. • Damping of the longitudinal emittance at the expense of the transverse damping. • 6 -D phase spare area “Robinson’s” Theorem • Numerical examples and comparison with simulations are in progress. KJK 2/5/02 IIT Cooling Theory/Simulation Day 21 Advanced Photon Source